The (p, q, r, l) model for stochastic demand under Intuitionistic fuzzy aggregation with Bonferroni mean

Abstract

This paper investigates a hill type economic production-inventory quantity (EPIQ) model with variable lead-time, order size and reorder point for uncertain demand. The average expected cost function is formulated by trading off costs of lead-time, inventory, lost sale and partial backordering. Due to the nature of the demand function, the frequent peak (maximum) and valley (minimum) of the expected cost function occur within a specific range of lead time. The aim of this paper is to search the lowest valley of all the valley points (minimum objective values) under fuzzy stochastic demand rate. We consider Intuitionistic fuzzy sets for the parameters and used Intuitionistic Fuzzy Aggregation Bonferroni mean for the defuzzification of the hill type EPIQ model. Finally, numerical examples and graphical illustrations are made to justify the model.

Appendices

Appendix 1

Some more definitions over IFBM (Xu and Yager 2011):

Definition 12

Let $A_{i}(i=1,2,\ldots ,n)$ be a set of nonnegative numbers and $p,q\ge 0$, then the Bonferroni mean is given by the following function.

$$\begin{aligned}&B^{p,q}(A_1,A_2,\ldots ,A_n)\nonumber \\&\quad =\left( \begin{array}{l}\frac{1}{n(n-1)}\sum _{i, j=1,i\ne j}^{n}A_{i}^{p}A_{j}^{q}\end{array}\right) ^{\frac{1}{p+q}} \end{aligned}$$

(62)

Definition 13

Let $A_{i}=\langle \mu _{A_i},\nu _{A_i}\rangle \epsilon ~ \texttt {IFNs}(X)(i=1,2,\ldots ,n)$ and $p,q\ge 0$, then Intuitionistic fuzzy interaction Bonferroni mean (IFIBM) is defined as

$$\begin{aligned}&{} \textit{IFIBM}^{p,q}(A_1,A_2,\ldots ,A_n)\nonumber \\&\quad =\left( \begin{array}{l}\frac{1}{n(n-1)} \bigoplus _{i, j=1,i\ne j}^{n}A_{i}^{p}\widehat{\bigotimes } A_{j}^{q}\end{array}\right) ^{\frac{1}{p+q}} \end{aligned}$$

(63)

where $A_{i}^{p}=\langle (1-\mu _{A_i})^{p}-(1-\mu _{A_i}-\nu _{A_i})^{p}, 1-(1-\nu _{A_i})^{p}\rangle$ and $A_{j}^{q}=\langle (1-\mu _{A_i})^{q}-(1-\mu _{A_i}-\nu _{A_i})^{q}, 1-(1-\nu _{A_i})^{q}\rangle$ by Eq. (29). Now,

$\textit{IFIBM}^{p,q}(A_1,A_2,\ldots ,A_n)=\langle (1-(\prod _{i,j=1,i\ne j}^{n}((1-(1-\nu _{A_i})^{p}(1-\nu _{A_j})^{q}+$ $(1-\nu _{A_{i}}-\nu _{A_{j}})^{p}(1-\mu _{A_{i}}-\nu _{A_{j}})^{q})))^{\frac{1}{n(n-1)}} +$ $(\prod _{i,j=1,i\ne j}^{n}((1-\mu _{A_i}-\nu _{A_j})^{q}(1-\mu _{A_i}-\nu _{A_j})^{q}))^{\frac{1}{n(n-1)}})^{\frac{1}{p+q}}-((\prod _{i,j=1,j\ne j}^{n}((1-\mu _{A_i}-\nu _{A_j})^{p}(1-\mu _{A_i}-\nu _{A_j})^{q}))^{\frac{1}{n(n-1)}})^{\frac{1}{p+q}},$ $1-(1-(\prod _{i,j=1,j\ne j}^{n}((1-(1-\nu _{A_i})^{p}(1-\nu _{A_j})^{q}+ (1-\nu _{A_i}-\nu _{A_j})^{p}(1-\mu _{A_i}-\nu _{A_j})^{q})))^{\frac{1}{n(n-1)}} +(\prod _{i,j=1,j\ne j}^{n}((1-\mu _{A_i}-\nu _{A_j})^{q}(1-\mu _{A_i}-\nu _{A_j})^{q}))^{\frac{1}{n(n-1)}})^{\frac{1}{p+q}}\rangle (A.2)$

Definition 14

Let $A_{i}=\langle \mu _{A_i},\nu _{A_i}\rangle \epsilon ~ \texttt {IFNs}(X)(i=1,2,\ldots ,n)$ and $p,q\ge 0, w=(w_{1}, w_2,\ldots ,w_n)^{T}$ is the weighting vector of $A_i$ with $w_i\epsilon [0,1]$ and $\sum _{i=1}^{n}w_i=1$, then the weighted Intuitionistic fuzzy interaction Bonferroni mean (WIFIBM) is defined as

$$\begin{aligned}&{\textit{WIFIBM}}^{p,q}(A_1,A_2,\ldots ,A_n)\\&\quad =\left( \begin{array}{c}\frac{1}{n(n-1)}\bigoplus _{i, j=1,i\ne j}^{n}(nw_i A_{i})^{p}\widehat{\bigotimes } (nw_j A_{j})^{q}\end{array}\right) ^{\frac{1}{p+q}} \end{aligned}$$

where $A_{i}^{p}=\langle (1-\mu _{A_i})^{p}-(1-\mu _{A_i}-\nu _{A_i})^{p},1-(1-\nu _{A_i})^{p}\rangle$ and $A_{j}^{q}=\langle (1-\mu _{A_i})^{q}-(1-\mu _{A_i}-\nu _{A_i})^{q},1-(1-\nu _{A_i})^{q}\rangle$ by Eq. (29)

Appendix 2

Here, we have introduced a LINGO 8.0 programming and its outputs: (We note that, if the value of several functions are zero then reset the functions with appropriate sign and proceed as usual)

• Step 1: MIN$=$Z;

• Step 2: Z2$=$F1*S1 $+$ F2*S2 $+$ F3*S3 $+$ F4*S4 $+$ F5*S5 $+$ F7*S7 $+$ F8*S8 $+$ F9*S9 $+$ F10*S10 $+$ F11*S11$+$

• Step 3: F12*S12 $+$ F13*S13 $+$ F14*S14 $+$ F;Z1$=$ F6*S6;Z$=$Z2 - Z1;

• Step 4: SM$=$60;LM$=$12;DEL$=$1000;GAMA$=$5;PI$=$8;PI0$=$20; E$=$0.001; M$=$100;MU$=$15;SIGMA$=$(5) ${^{\wedge }}$ 0.5;K$=$1.4;

• Step 5: S1$=$(6*A*R*L*L - 3*B*L*R*R $+$ 2*C*R$^{\wedge }$3)/(6*L $^{\wedge }$3);

• Step 6: S2$=$(6*A*R*R*L*L - 4*B*L*R$^{\wedge }$3 $+$ 3*C*R$^{\wedge }$4)/(12*L$^{\wedge }$4);

• Step 7: S3$=$(6*A*L*L*(P*L - R) - 3*B*L*(P$^{\wedge }$2*L$^{\wedge }$2 - R$^{\wedge }$2) $+$ 2*C*(P$^{\wedge }$3*L$^{\wedge }$3 - R$^{\wedge }$3))/(6*L$^{\wedge }$3);

• Step 8: S4$=$(6*A*L*L*(P$^{\wedge }$2*L$^{\wedge }$2 - R$^{\wedge }$2) - 4*B*L*(P$^{\wedge }$3*L$^{\wedge }$3 - R$^{\wedge }$3) $+$ 3*C*(P$^{\wedge }$4*L$^{\wedge }$4 - R$^{\wedge }$4))/(12*L$^{\wedge }$4);

• Step 9: S5$=$A* @ log(P*L/R) - B*(P - R/L) $+$ 0.5*C*(P$^{\wedge }$2*L$^{\wedge }$2 - R$^{\wedge }$2)/(L$^{\wedge }$2);

• Step 10: S6$=$ - (P*P*C - P*B $+$ A)*@log((E*L)/(P*L - R)) $+$ (2*P*C - B)*(P - R/L) - 0.5*C*(P*L - R)$^{\wedge }$2/(L$^{\wedge }$2);

• Step 11: S7$=$(6*A*P - 3*B*P$^{\wedge }$2 $+$ 2*C*P$^{\wedge }$3)/6;S8$=$(6*A*P$^{\wedge }$2 - 4*B*P$^{\wedge }$3 $+$ 3*C*P$^{\wedge }$4)/12;

• Step 12: S9$=$A*@log(P/E) - B*P $+$ 0.5*C*P*P;

• Step 13: S10$=$(- P*P*C $+$ P*B - A)*@log(E/P) $+$ (- 2*P*C $+$ B)*P $+$ 0.5*C*P*P;

• Step 14: S11$=$(6*A*(M-P) - 3*B*(M$^{\wedge }$2 - P$^{\wedge }$2) $+$ 2*C*(M$^{\wedge }$3 - P$^{\wedge }$3))/6;

• Step 15: S12$=$(6*A*(M$^{\wedge }$2 - P$^{\wedge }$2) - 4*B*(M$^{\wedge }$3 - P$^{\wedge }$3) $+$ 3*C*(M$^{\wedge }$4 - P$^{\wedge }$4))/12;

• Step 16: S13$=$A*@log(M/P) - B*(M-P) $+$ 0.5*C*(M$^{\wedge }$2 - P$^{\wedge }$2);

• Step 17: S14$=$(P*P*C - P*B $+$ A)*@log((M - P)/E) $+$ (2*P*C - B)*(M - P) $+$ 0.5*C*(M - P)$^{\wedge }$2;

• Step 18: A$=$3*(10*MU$^{\wedge }$2 $+$ 10*SIGMA$^{\wedge }$2 $+$ 3*M$^{\wedge }$2 - 12*MU*M)/(M$^{\wedge }$3);

• Step 19: B$=$12(15*MU$^{\wedge }$2 $+$ 15*SIGMA$^{\wedge }$2 $+$ 3*M$^{\wedge }$2 - 16*MU*M)/(M$^{\wedge }$4);

• Step 20: C$=$30*(6*MU$^{\wedge }$2 $+$ 6*SIGMA$^{\wedge }$2 $+$ M$^{\wedge }$2 - 6*MU*M)/(M$^{\wedge }$5); L$=$0.1;

• Step 21: D$=$(6*A*M$^{\wedge }$2 - 4*B*M$^{\wedge }$3 $+$ 3*C*M$^{\wedge }$4)/12;R$=$ MU*L $+$ K*SIGMA*L$^{\wedge }$0.5;

• Step 22: F1$=$LM*R*L;F2$=$0.5*LM*L*L;F3$=$0.5*PI*L*(L*P - 4*R);F4$=$PI*L*L;F5$=$0.5*(LM $+$ PI)*R*R;

• Step 23: F6$=$0.5*PI*(L*P - R)$^{\wedge }$2;F7$=$0.5*LM*L*(L*P - 2*R);F8$=$F2;

• Step 23: F9$=$0.5*LM*((Q $+$ R)$^{\wedge }$2 - R*R);F10$=$0.5*LM*((Q $+$ R)$^{\wedge }$2 $+$ 2*R*L*P-L$^{\wedge }$2*P$^{\wedge }$2);

• Step 24: F11$=$0.5*(PI $+$ PI0)*L*(4*SM $+$ 3*L*P - 4*R);F12$=$2*L*L*(PI $+$ PI0);

• Step 25: F13$=$0.5*R*R*(LM $+$ PI $+$ PI0);

• Step 26: F14$=$0.5*(PI $+$ PI0)*(SM*SM $+$ 4*L*P*SM $+$ 3*P*P*L*L - 2*R*SM - 2*R*L*P);

• Step 27: F$=$DEL*(GAMA)$^{\wedge }$( - L);

• Local optimal solution found at iteration: 262

• Objective value: 965.2966

• ${\text {Z}} = 965.2966$, ${\text {P}}= 99.99900$, ${\text {D}}= 175.6337$, ${\text {Q}} =4.595212$, ${\text {R}}=2.489949$, ${\text {F1}} = 2.987939$, ${\text {S1}}=0.7526193$, ${\text {F2}} =0.6000000{\text {E}}\!-\!01$, ${\text {S2}}=8.144608$, ${\text {F3}} =0.1604087{\text {E}}\!-\!01$, ${\text {S3}}=0.2473798$, ${\text {F4}}=0.8000000{\text {E}}\!-\!01$; $S4=6.855310$, ${\text {F5}}=61.99848$, ${\text {S5}}=0.7992532{\text {E}}\!-\!02$, ${\text {F6}}=225.5974$, ${\text {S6}}=0.1088008{\text {E}}\!-\!01$, ${\text {F7}}=3.012001$, $S7=0.9999992$, ${\text {F8}}=0.6000000{\text {E}}\!-\!01$, ${\text {S8}}=14.99992$, ${\text {F9}}= 263.9980$, ${\text {S9}}=0.4154986$, ${\text {F10}} =0.4314534$, ${\text {S10}}=0.1595342{\text {E}}\!-\!01$, ${\text {F11}}=364.0559$, S11=0.8156408E-06,F12=0.5600000 ${\text {S12}}=0.8156367{\text {E}}\!-\!04$, ${\text {F13}}=123.9970$, ${\text {S13}}=0.8156449{\text {E}}\!-\!08$, ${\text {F14}}= 83319.29$, ${\text {S14}}=0.00$, ${\text {F}}=851.3399$, ${\text {A}} =0.4281708{\text {E}}\!-\!01$, ${\text {B}}=0.1129025{\text {E}}\!-\!02$, ${\text {C}} =0.7090249{\text {E}}\!-\!05$

Appendix 3

Convergent Analysis of the proposed Algorithm

• Step 1: The crisp model is convergent because the parameters associated to it are bounded with some practical assumptions.

• Step 2: In the assumptions of triangular membership and non-membership functions for the parameters, it is noted that they must intersect at some points. In our study we have the member and non-member of the parameter r as follows: $\mu _{r}(r) =\left\{ \begin{array}{ll}\frac{r-r_L}{r_R-r_L} &{}\;\; \texttt {for}~ r_L\le r\le r_C\\ \frac{r_R-r}{r_R-r_C} &{}\;\; \texttt {for}~ r_C\le r\le r_R\\ 0 &{}\;\; \texttt {for} ~~\texttt {elsewhere}\end{array}\right\}$ and $\nu _{r}(r)=\left\{ \begin{array}{ll}\frac{r_C-r}{r_C-r_{L}^{\prime }} &{}\;\; \texttt {for}~ r_{L}^{\prime }\le r\le r_C\\ \frac{r -r_C}{r_{R}^{\prime }-r_C} &{}\;\; \texttt {for}~ r_C\le r\le r_{R}^{\prime }\\ 0 &{}\;\; \texttt {for} ~~\texttt {elsewhere}\end{array}\right\}$ respectively. To get an intersection we write: $\frac{r-r_L}{r_R-r_L}=\frac{r_C-r}{r_C-r_{L}^{\prime }}\rightarrow r =\frac{r_C r_Rr_L r_{L}^{\prime }}{r_C+r_R-r_L-r_{L}^{\prime }}$ and $\frac{r_R-r}{r_R-r_C}=\frac{r-r_C}{r_{R}^{\prime } - r_C }\rightarrow r=\frac{r_R r_{R}^{\prime }-r_{C}^{2}}{r_R + r_{R}^{\prime }-2r_C}$.So the convergent region is the interval $[ \frac{r_R r_{C}-r_{L}r_{L}^{\prime }}{r_R+ r_{C}-r_L -r_{L}^{\prime }},\frac{r_R r_{R}^{\prime }-r_{C}^{2}}{r_R + r_{R}^{\prime }-2r_C}]$. As $r_{R}^{\prime }\rightarrow r_R$ and $r_{L}^{\prime }\rightarrow r_L$, this interval reduces to $[ \frac{r_R r_{C}-r_{L}^{2}}{r_R+ r_{C}-2r_L},\frac{ r_{R}^{2}-r_{C}^{2}}{2r_R-2r_C}]$. Again, as $r_R\rightarrow r_C\leftarrow r_L$, this interval reduces to $[ \frac{r_R r_{C}-r_{C}^{2}}{2(r_R-r_C)},\frac{ r_{R}^{2}-r_{C}^{2}}{2(r_R-r_C)}]\Rightarrow r\rightarrow {r_C}$. Hence r is convergent. Similarly, we find for the other parameters also.

• Step 3: The probability density function of the demand rate is given by $f(x)=\left\{ \begin{array}{cc} a-bx+cx^{2}, &{} 0\le x\le M\\ 0, &{} elsewhere \end{array}\right\}$ where $a=\frac{3}{M^{3}}[10\mu ^{2}+10\sigma ^{2}+3M^{2}-12\mu M], b=\frac{12}{M^{4}}[15\mu ^{2}+15\sigma ^{2}+3M^{2}-16\mu M]$, $c=\frac{30}{M^{5}}[6\mu ^{2}+6\sigma ^{2}+M^{2}-6\upmu M], \mu$ is the mean and $\sigma$ is the standard deviation of the given distribution respectively. For fuzzy environment we can assume the probability density function of the demand rate as $\phi (x)=\left\{ \begin{array}{cc} a+bx-cx^{2}, &{} 0\le x\le M\\ 0, &{} elsewhere \end{array}\right\}$. So the intersections between f(x) and $\phi (x)$ are given by $a-b x+cx^{2}=a+bx-cx^{2}\Rightarrow bx=cx^{2}\Rightarrow x=0, b/c$. Thus, two density functions must intersect at least one point. Hence it is convergent.

• Step 4: All the parameters associated with the fuzzy model have lower and upper bounds. Therefore, the objective function has upper and lower bounds by its virtue. If L and U are the lower and upper bounds of the objective function then, as per assumptions, the membership and non membership functions of the objective function is given by $\mu (Z)=\left\{ \begin{array}{ll}1 &{} \quad \texttt {if}~ Z\le L\\ \theta _{1}e^{-\frac{1}{2}\big (\frac{Z-L}{U-L}\big )^{2}} &{} \quad \texttt {if}~ L\le Z\le U\\ 0 &{} \quad \texttt {if} Z\ge U\end{array}\right\}$ and $\nu (Z) =\left\{ \begin{array}{ll}1 &{} \quad \texttt {if}~ Z\le L\\ \theta _{2}\sqrt{\big (\frac{U-L}{2}\big )^{2}-\big (Z-\frac{U+L}{2}\big )^{2}} &{} \quad \texttt {if}~ L\le Z\le U\\ 1 &{} \quad \texttt {if} Z\ge U\end{array}\right\}$ respectively. Now to get an intersection we must write $\theta _{1} e^{-\frac{1}{2}\big (\frac{Z-L}{U-L}\big )^{2}}=\theta _{2}\sqrt{ \big (\frac{U-L}{2}\big )^{2}-\big (Z-\frac{U+L}{2}\big )^{2}}\Rightarrow \big (\frac{Z-L}{U-L}\big )^{2}=\ln |\frac{\theta _1}{\theta _2}|-\ln | \big (\frac{U-L}{2}\big )^{2}- \big (Z-\frac{U+L}{2}\big )^{2}|\Rightarrow Z^2-2LZ+(U-L)^2 \ln |Z(U+L)-Z^2 -UL| +L^2-(U-L)^2 \ln |\frac{\theta _1}{\theta _2}|=0$. The above equation has at least one root because, as $U\rightarrow L$ the above implies $Z^2 -2LZ+L^2=0\Rightarrow Z=L$. Hence the objective function is convergent.

• Step 5: Since the model has been solved under three cases namely, (i) $S(z)< BM$, (ii) $S(z)=BM$ and (iii) $S(z)>BM$ so the score values of the objective function must be bounded below, around and above the Bonferroni mean (BM) respectively. Hence the solutions are convergent.